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If the space happens to also be a [[normed space]] (or a [[seminormed space]]), such as the scalar field with the [[absolute value]] for instance, then a subset <math>S</math> is von Neumann bounded if and only if it is [[Norm (mathematics)|norm]] bounded; that is, if and only if <math>\sup_{s \in S} \|s\| < \infty.</math>
If <math>S \subseteq X</math> is a set then <math>F : X \to Y</math> is said to be {{em|bounded on <math>S</math>}} if <math>F(S)</math> is a [[Bounded set (topological vector space)|bounded subset]] of <math>Y,</math> which if <math>(Y, \|\cdot\|)</math> is a normed (or seminormed) space happens if and only if <math>\sup_{s \in S} \|F(s)\| < \infty.</math>
A linear map <math>F</math> is bounded on a set <math>S</math> if and only if it is bounded on <math>x + S</math> for every <math>x \in X</math> (because <math>F(x + S) = F(x) + F(S)</math> and any translation of a bounded set is again bounded).
By definition, a linear map <math>F : X \to Y</math> between [[Topological vector space|TVS]]s is said to be {{em|[[Bounded linear operator|bounded]]}} and is called a {{em|[[bounded linear operator]]}} if for every [[Bounded set (topological vector space)|(von Neumann) bounded subset]] <math>B \subseteq X</math> of its ___domain, <math>F(B)</math> is a bounded subset of it codomain; or said more briefly, if it is bounded on every bounded subset of its ___domain. When the ___domain <math>X</math> is a normed (or seminormed) space then it suffices to check this condition for the open or closed unit ball centered at the origin. Explicitly, if <math>B_1</math> denotes this ball then <math>F : X \to Y</math> is a bounded linear operator if and only if <math>F\left(B_1\right)</math> is a bounded subset of <math>Y;</math> if <math>Y</math> is also a (semi)normed space then this happens if and only if the [[operator norm]] <math>\|F\| := \sup_{\|x\| \leq 1} \|F(x)\| < \infty</math> is finite.
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